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Graphs and Combinatorics 6, 1-4 (1990) Combinatorics

Summary: Graphs and Combinatorics 6, 1-4 (1990)
Transversal Numbers of Uniform Hypergraphs
Noga Alon*
Department of Mathematics, Sackler Faculty of Exact Sciences,Tel AvivUniversity,Tel Aviv,
Israel, and BellCommunications Research, Morristown, NJ 07960,USA
Abstract. The transversal number ~(H)of a hypergraph H is the minimum eardinality of a set of
verticesthat intersects all edgesofH. For k ~ 1defineck= supz(H)/(ra+ n),where H ranges over
all k-uniformhypergraphs with n verticesand medges.Applyingprobabilistic arguments we show
that ck= (1 + o(1))~,r. This settles a problem ofTuza.
t ~
1. Introduction
The transversal number ~(H) of a hypergraph H is the minimum cardinality of a set
of vertices that intersects all edges of H. Let H = (V,E) be a k-uniform hypergraph
with n vertices and m edges. Trivially, there exists a positive constant cg (which
depends only on k) such that z(H) <_ck(n + m). Tuza [4] proposed the problem of
determining or estimating the best possible constants Ckwith the above property.
Clearly these constants are given by c~ = sup z(H)/(m + n), where H ranges over all


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics